A Banach space with $L$-orthogonal sequences but without $L$-orthogonal elements

TL;DR

The paper addresses the problem of whether there exists a Banach space with an $L$-orthogonal sequence but without an $L$-orthogonal element in its bidual, and shows this phenomenon is independent of ZFC by introducing $Q$-measures as a measure-theoretic generalization of $Q$-points. It develops a framework linking set-theoretic invariants, forcing models (including Laver/Miller), and measure extensions to $Q^+(\,\omega\)$-measures, strong $Q$-measures, and fit $Q$-measures. The authors establish equivalences between these notions and invariants like $\mathfrak{d}$ and $cov(\mathcal{M})$, prove consistency of no $Q$-measures in established models, and construct Banach spaces that witness the independence in both directions. They culminate with a concrete space built from non-fit $Q$-measures that contains an $L$-orthogonal sequence but no $L$-orthogonal elements in $X^{**}$, while also outlining open questions about the relationships among the various $Q$-measure notions and their impact on $L$-orthogonality.

Abstract

We prove that the existence of Banach spaces with $L$-orthogonal sequences but without $L$-orthogonal elements is independent of the standard foundation of Mathematics, ZFC. This provides a definitive answer to \cite[Question~1.1]{AvilesMartinezRueda}. Generalizing classical $Q$-point ultrafilters, we introduce the notion of $Q$-measures and provide several results generalizing former theorems by Miller \cite{Miller} and Bartoszynski \cite{Bartoszynski} for $Q$-point ultrafilters.

Theorems & Definitions (59)

• Theorem 1: Independence of $L$-elements
• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Theorem 3.1
• Lemma 3.2
• proof